Quantitative extension of Dickson's prime k-tuples conjecture

A finite set of integers $\{a_1,a_2,\ldots,a_k\}$ is admissible if, for every prime $q$, its residue classes do not cover all residue classes modulo $q$. Quantitative extension of Dickson's conjecture. For every arbitrarily small constant $\delta>0$, there exists an integer $x_0=x_0(\delta)$ such that, for every $x\geq x_0$, if $a_1,a_2,\ldots,a_k$ is an admissible set with

$$k\leq (\log x)^{1-\delta},\qquad a_i\leq x^{1-\delta},$$

then there is a positive integer $n\leq x$ such that all of

$$n+a_1,n+a_2,\ldots,n+a_k$$

are prime. The source presents this as a reasonable conjecture extending Dickson's conjecture; no resolution is given.